quantitative prediction of the effects of mistuning ...quantitative prediction of the effects of...
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Quantitative Prediction of the Effects of Mistuning Arrangement on Resonant Response of aPractical Turbine Bladed Disc
Dr. E. Petrov, Dr. K. Sanliturk, Prof. D. EwinsImperial College of Science, Technology and Medicine
Mechanical Engineering Department, Centre of Vibration EngineeringLondon, SW7 2BX, UK.
R. ElliottTurbine Systems - EngineeringRolls-Royce Plc, PO Box 31
Derby, DE24 8BJ, UK.
Abstract
The effects of blade mistuning have been studiedextensively in the past although most of theresearch efforts have been directed towardsunderstanding its qualitative aspects. Addressingthe quantitative aspects of the mistuning problem,especially for industrial designs, has beensomewhat limited in the past, mainly due to theneed for using reduced-order and thus simplifiedmodels in such analyses. In other words, it has beendifficult to translate previous mistuning resultsdirectly for high-cycle fatigue predictions for agiven bladed-disc.
A new method for the dynamic analysis ofmistuned bladed discs is presented. The method isbased on exact calculation of the response of themistuned system using response levels of the tunedassembly and a modification matrix constructedfrom the Frequency Response Function (FRF)matrix of the tuned system and a matrix describingthe mistuning. The main advantages of the methodare its efficiency and accuracy, which allows theuse of large finite element models of practicalbladed disc assemblies in parametric studies ofmistuning effects on vibration amplitudes. A newmethod of calculating the FRF matrix of the tunedsystem using a sector model is also developed so asto improve the efficiency of the method further,making the proposed method a very attractive toolfor mistuning studies of practical design.
1. Introduction
Small variations of individual blade characteristicsin a bladed assembly inevitably arise during blademanufacturing and assembling processes. It is wellknown that the dynamic response of a mistunedbladed disc can be altered significantly from that ofits tuned counterpart so that the response ofamplitudes of individual blades may vary widelywithin the same assembly (Dye and Henry, 1969),(Ewins, 1969) and (Whitehead, 1966). This
situation has posed a very important practicalproblem for a long time and efforts of manyinvestigators have been, and continue to be,devoted to predicting and controlling thesemistuning effects. Surveys of recent studies ofmistuning are given in (Ewins, 1991) and (Slater,1999).
The current trend in the turbomachineryindustry is to avoid the various simplifyingassumptions during the modelling process and toseek quantitative answers to questions related tomistuning. Using realistic finite element models forthe complete bladed disc, including mistuningeffects is, however, still too expensive in spite ofthe advances in finite element modelling andcomputer hardware during recent years. Thecommon practice in industry is to make use of thecyclic symmetry properties of the tuned system toobtain the natural frequencies and mode shapes ofthe whole tuned bladed disc assemblies. Althoughvery desirable, there is no method readily availableto use the models based on cyclic symmetry formistuning studies. Attempts at using detailed finiteelement models for mistuned bladed discs havebeen made recently. An original approach has beendeveloped in (Bladh et al., 1998) and (Castanier etal. 1997) to reduce the size of the large-scale finiteelement models of mistuned bladed discs. Thedeveloped technique is similar to the Ritzprocedure and is based on expansion of themistuned bladed disc amplitudes into a series ofmode shapes calculated for two specially chosensubsystems of the tuned bladed disc. Someinvestigation of the validity of this technique can befound in reference (Frey, 1998). A different methodis proposed in (Yang and Griffin, 1998), wheremodel reduction is carried out by representation ofmode shapes as a limited sum of the system modesthat are called “nominal” modes by the authors.These can be the modes of the tuned bladed disc orof the bladed disc with a specified mistuningpattern. Both of the above mentioned approaches
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are dependent on the nominal mode shapes chosenfor the expansions of the mistuned system responseas well as on the number of these mode shapes.
The present paper introduces a new method forthe analysis of forced vibration of mistuned bladeddiscs. The method is based on an exact relationshipbetween the response levels of the tuned and themistuned bladed discs, the tuned system beingdescribed by a sector model using the cyclicsymmetry properties of the assembly. An exactrelationship between the tuned and the mistunedsystem is derived using Woodbury-Sherman-Morrison formula for the inverse of a perturbedmatrix (Sherman and Morrison, 1949) and(Woodbury, 1950). This formula was applied inreference (Level et al., 1996) for responsereanalysis of a simple linear system and wasproposed and used successfully for calculations ofmodifications in the non-linear analysis of systemswith friction dampers in references (Sanliturk et al.,1999). An important feature of the methodproposed here is the reduction of the system modelto a manageable size without introducing any lossof accuracy during the reduction process, such as isusually incurred when including only a subset ofthe model degrees of freedom reflecting the interestof the eventual application. Any subset of nodes ofinterest can be chosen from the whole set ofassembly nodes for the mistuning analysis withoutany adverse effect. Accordingly, the method allowsmistuning analysis of bladed discs using large finiteelement models that are used at present only in theanalysis of tuned bladed discs based on cyclicsymmetry approach.
The second section of the paper describes anexact relationship between the response amplitudesof the mistuned and the tuned bladed discs as wellas how to perform the response calculations for themistuned system at those co-ordinates where thesystem is mistuned or amplitudes are controlled.The section explores also how the tuned system canbe used during the forced response analysis of amistuned assembly. An original technique ispresented for the calculation of both forcedresponse levels and the Frequency ResponseFunction (FRF) matrix of tuned bladed discs usinga sector model. This method allows one to obtain atuned assembly’s response properties underarbitrarily distributed harmonic loads over allbladed disc sectors, a feature essential for theanalysis of mistuned bladed discs. Mistuningdevices or ‘elements’ are defined in the thirdsection to facilitate simulation of the differentmistuning conditions encountered in practice. Anapproach to establish a relationship betweenmistuning elements and blade-alone frequency
mistuning is also described. The last section of thepaper presents a case study of a mistuned turbinebladed disc with mistuned blades and withdamaged blades which have very large deviationsin blade-alone natural frequencies.
2. A new method for forced responseanalysis of mistuned bladed discs
2.1 An exact relationship between the responseof tuned and mistuned bladed discs
The equation of motion for forced vibration of abladed disc such as that shown in Fig.1 can bewritten in a customary form in the frequencydomain as:
fqZqDMK =ω=+ω− )()( 2 i (1)
where q is a vector of complex response amplitudesfor nodal displacements; f is a vector of complexamplitudes of harmonic nodal loads; K, M and Dare stiffness, mass and structural damping matricesof the system, respectively; )(ωZ is the dynamicstiffness matrix; ω is excitation frequency; and
1−=i . One can also include other termsrepresenting gyroscopic and stiffening effects dueto rotation in the dynamic stiffness matrix, ifrequired.
It is proposed here to represent the dynamicstiffness matrix of a mistuned bladed disc as a sumof two matrices; a matrix corresponding to thetuned bladed disc, 0Z , and a mistuning matrix,
Z∆ , which reflects the deviation from the tunedsystem. As a result, the matrix equation for theforced response of a mistuned bladed disc (1) canbe written as:
Fig. 1 Finite element model of a bladed disc
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[ ] fqZZ =ω∆+ω )()(0 (2)
and so the response of the mistuned bladed disc isexpressed as:
[ ] fZZq 10 )()( −ω∆+ω= (3)
However, calculation of the response levels bydirect solution of Eq.(2), or using the inverse of amatrix as implied by the above equation, isextremely costly, especially for the analysis of acomplete mistuned bladed disc assembly when arealistic finite element model is to be used. Themajor idea in this paper is to obtain the forcedresponse levels for a mistuned bladed disc withhigh accuracy without the need for:
(i) a matrix inversion;(ii) a complete finite element model for thewhole assembly,(iii) inclusion of all the degrees of freedom inthe response analysis.
It is shown in references (Sherman andMorrison, 1949) and (Woodbury, 1950) that theinverse of the sum of two matrices may be obtainedusing the so-called Sherman-Morrison-Woodburyidentity which relates the inverse of the sum of twomatrices to the inverse of one of the summands,transformed with the use of the second summand:
10
110
10
10
10 )()( −−−−−− +−=+ ZVUZVIUZZUVZ TTT (4)
This identity is valid for any matrix Z0 (N×N) andany matrix ∆Z= TUV , expressed by multiplicationof two rectangular matrices U (N×n) and V (N×n) ifZ0 and UZVI 1
0−+ T are invertible. Here I is an
identity matrix, N is the total number of degrees offreedom for the system considered, and n is numberof degrees of freedom where modifications aremade and forced response analysed.
For the mistuning problem addressed in thispaper, the matrix 0Z is the dynamic stiffnessmatrix of the tuned bladed disc and the matrix Z∆represents the “mistuning matrix” which can itselfbe represented as a multipli cation of two matrices.This representation can be done in various ways,one of which is to collect all nonzero rows into amatrix V T (n×N) and to construct the matrix U(N×n) using unit vectors corresponding to nonzerorows of the matrix V. Inserting Eq.(4) into Eq.(3),one can obtain an expression for the response of themistuned bladed disc in terms of that of the tunedsystem and the modification matrices U and V as
[ ][ ] 0
1000
01
000
)(
)(
qVUAVIUAq
fAVUAVIUAAqTT
TT
−
−
+−=
=+−=(5)
where [ ] 100 )()( −ω=ω ZA is the receptance (FRF)
matrix for the tuned bladed disc, and 0q is a vectorrepresenting the amplitudes of the tuned assemblydue to the external force vector, f.
An effective method for calculation the FRFmatrix of the tuned bladed disc is presented inreference by Petrov et al. [8, 9], where naturalfrequencies and mode shapes obtained from asector model (Fig.2a) of the tuned assembly areused for generation of the FRF matrix for the wholeassembly.
a)
b)
Fig. 2 Finite element sector model (a) and activenodes (b)
The expression for the FRF matrix of the wholebladed-disc assembly derived in reference [10] hasthe following form:
4
=
−
−
11
112
21
0
...
............
...
...
HHH
HHH
HHH
A
TN
TN
NT
N
BB
B
B
(6)
where
/ 2( 1) ( )
/ 2
1 α− −
=−
= ∑B
B
Ni k j k
j k Sk NB
c eN
H A (7)
and BN is a number of blades in the assembly;
2 /α π= BN ; 1=kc for all k, except 0.5=kc for
/ 2= Bk N when BN is even. ( )kSA are so-called
‘wave’ FRF matrices corresponding to forwardtravelling (wave number +k) and backwardtravelling (wave number -k) waves of engine orderexcitation. They can be can be expressed throughthe complex natural modes of one sector only as:
forward travelling wave
( )( )∑
= ω−ωη+φφ=ω
m
rk
rr
Tkr
krk
Si1
22)(
)()()(
)1()(A ; (8)
back travelling wave
( )( )∑
=
−
ω−ωη+φφ=ω
m
rk
rr
Tkr
krk
Si1
22)(
)()()(
)1()(A (9)
where m is a number of modes used for the FRFmatrix generation; ( )ω k
r is a natural frequency forthe r-th mode and k-th wave (or cyclic index)number; rη is the damping loss factor;
Im)(Re)()( kr
kr
kr iφ+φ=φ are mass normalised mode
shapes, which are complex for all k except0 and / 2Bk N= ; and a line over a symbol denotes
complex conjugation. It should be noted that thematrices ( )k
SA and ( )−kSA are not Hermitian
conjugates due to the presence of damping in theconsidered system.
It should be noted that Eq.(5) is an exactexpression and one of its very useful properties isthe possibility it offers of obtaining the response ofa mistuned system by considering only a smallsubset of the degrees of freedom. These degrees offreedom are: (i) those where the mistuning isapplied and (ii) those where the forced responselevels are of interest, and this combination isreferred to below as the active co-ordinates.
The distribution of forces over the blade nodesare taken into account when forced response thetuned assembly is calculated, which can be doneefficiently even for the case when the loads areapplied to large number of nodes. Owing to
accounting for the forces at the stage of tunedsystem calculation, the nodes where the forces areapplied may not be included into the set the activenodes. An example of active nodes (shown by redcircles) and nodes where forces are applied (shownby green circles) is given in Fig.2b.
As a result of these properties, the size of thematrices can be reduced to any desired levelwithout any loss of exactness of the description ofthe behaviour. This can be achieved as follows.First, the vector of complex displacementamplitudes is partitioned into two: the part with theactive co-ordinates (index a) and the partcontaining all the other, passive, degrees of
freedom (index p), i.e. { }Tpa qqq ,= . The
modification, or mistuning, matrix, Z∆ , is alsopartitioned accordingly, as illustrated below.
T
aaT
aa
pppa
apaa
=
=
∆∆∆∆
=∆0
V
0
U
00
0VUzz
zzZ (10)
Then, by also partitioning the receptance matrix forthe tuned system 0A in the same way, theexpression in Eq.(5) can be written in the followingform:
0
10
00
00
000
0)(
+
−
=
−
p
aT
aaaaT
aa
pppa
apaa
p
a
p
a
q
qVUAVIU
AA
AA
q
q
q
q
{ } { }0 0 1
0 0 1 00
( )( )
−
− += − +
T Ta aa a a aa a a
T T ap pa a a aa a a
q A U I V A U V qq A U I V A U V (11)
where 0aaA , 0
apA , 0paA , 0
ppA are partitions of the
matrix, 0A , according to the unmodified andmodified co-ordinates. An important feature ofEq.(11) is that it expresses the response of themistuned system as that of the tuned system minusan expression which depends on (i) the FRF matrixof the tuned system, (ii) the mistuning matrix and(iii) the response of the tuned system at the activeco-ordinates. As a result, this feature allowscalculation of the mistuned response levels at theactive co-ordinates by considering only those rowsin Eq.(11) which correspond to the activecoordinates. The formulation in the next sectiondeals with active coordinates only, hence the indexa is dropped in subsequent development of theformulation.
2.2 Recurrence update of the forced response
It is seen that Eq.(11) needs the inversion of asquare matrix of order n which is the number of co-ordinates involved in the mistuning modifications.This matrix inversion process can also be avoidedby decomposing the modification matrix into a sum
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of individual modifications, ∑=
∆=∆n
jj
1
zZ , such that
each can be written as a multiplication of twovectors, i.e. T
jjjz vu=∆ . This yields a much simpler
formula for calculating the forced response levelsof mistuned systems. If the same scheme of matrixdecomposition as was illustrated in Fig.2 is adoptedhere, then u will be a unit vector with one non-zerocomponent and Eq.(5) will be reduced to a simpleexpression of the form:
jj
Tj
Tj A
Av
qvqq
+−=
10
0 (12)
where jA is the j-th column of matrix A0, and the
coefficient before Aj in Eq.(12) is a scalar value. Asa result, it is possible to take into account individualrows of the mistuning matrix sequentially, using thefollowing recurrence formula:
)()(
)()()1(
1j
jjj
Tj
jTjjj A
Av
qvqq
+−=+ (13)
( ))(
)()()()1(
1 jj
Tj
jTj
jjjj
Av
AvAAA
+−=+ (14)
To start the recurrence update, the responseamplitudes and the FRF matrix of the tuned bladeddisc are used.
3. Mistuning modelling
3.1 Mistuning elements
The method for the analysis for mistuned systemsproposed does not make any assumptions regardingthe distribution or the magnitude of the mistuningmatrix, Z∆ . However, it is useful to define,without any loss of generality, some simplemistuning elements that can be applied at desiredlocations in the bladed disc model and havephysical interpretation. Examples of such elementsare lumped mass, stiffness and damping mistuning
and the mistuning element matrix, ez∆ , can be
expressed as: mass mistuning
ω−ω−
ω−=∆
z
y
x
e
m
m
m
2
2
2
00
00
00
z (15)
stiffness mistuning damping mistuning
=∆
z
y
xe
k
k
k
00
00
00
z ;
=∆
z
y
xe
id
id
id
00
00
00
z (6)
where jm , jk ; jd are mass, stiffness and damping
coefficients of the element in three orthogonaldirections. Their combination allows us to describea wide range of mistuning patterns that might beencountered in practice.
3.2 Blade frequency mistuning
It is possible to describe quite a wide class ofpossible mistuning arrangements using the simplemistuning elements introduced in the previoussection. One of the widely-used measures ofmistuning in practice is to refer to the ‘blade-alone’cantilever-natural-frequency scatter. This type ofmistuning can easily be represented by establishinga relationship between individual blade frequenciesand the properties of a set of mistuning elementsapplied to the blades. This can be achieved byconducting a preliminary analysis to determine aone-to-one relationship between the blade alonefrequency scatter and the amount of mistuningintroduced at specific locations. A possibleapproach is to express the mistuning matrix for ablade, Z∆ , as
0ZZ ∆=∆ fc (17)
where 0Z∆ is a predefined mistuning matrix for agiven set of mistuning element characteristics andcf is a variable representing mistuning coefficient.Calculations are made for a sufficient number ofvalues of cf so as to establish a relationship with anacceptable accuracy between the mistuningcoefficient, cf, and the scatter in blade-alonefrequency. This relationship is usually non-linearbut a very good description can be made using aspline approximation. An example demonstratingsuch relationship is shown in Fig.3 for the firstblade-alone natural frequency when lumped massmistuning elements are applied to active nodesshown in Fig.2b.
It should be stated that some preliminaryblade-alone analysis may be necessary to determinethe datum mistuning matrix, 0Z∆ , especially if the
blade-alone frequency mistuning for more than onemode of vibration is of interest.
6
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-10.0
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Fig. 3 Relationship between mistuning coefficientand frequency mistuning
4. Numerical results
4.1 The bladed disc analysed and excitationconditions
The method developed above has beenimplemented in FORTRAN program and has beenapplied to the response analysis of a number ofrealistic mistuned bladed discs.
In the present section, an example of themethod applied to the analysis of a high-pressureturbine bladed disc shown in Fig.1 is represented.The bladed disc analysed comprises 92 blades. Itsfinite element single-sector model shown in Fig.2acontains 162,708 degrees of freedom (DOFs) andthe full bladed disc comprises about 15 millionsDOFs. The model describes, in detail, thegeometric shape of the blade and the disc, theblade-disc interface and the anisotropy of the bladematerial. The sector model is used fordetermination the first 16 natural frequencies andmode shapes of the tuned bladed-disc assembly forall their cyclic indices (nodal diameters) that arepossible for the considered assembly in the rangefrom 0 to 46. These modes of the tuned system areused for generating the FRF matrix of the tunedassembly.
A relatively small subset of nodes was selectedfor analysis of the mistuned system. The nodesused are shown in Fig.2b, where green circlesindicate the nodes subjected to loads and red circlesare the active nodes where response amplitudes arecalculated and condensed mass mistuning elementsare applied. The blade mistuning is assumedrandom and is normally distributed in the blade setmanufactured. Realistic characteristics of thenormal distributions (i.e. the mean value and thestandard deviation) for first four blade-alone natural
frequencies (namely, first flap, 1F, and firsttorsional, 1T) are determined as a result ofstatistical analysis of several hundreds of theexperimental values. The mistuning patternsanalysed are then generated using a random numbergenerator for the obtained distributioncharacteristics.
Moreover, the influence on response levels ofdamaged blades, which can have large mistuninglevels, is analysed. In the cases analysed here, thedamaged blades can reach 34% discrepancy in thefirst natural frequency compared with thecorresponding value for the tuned blade. In themistuning patterns analysed, damaged blademistuning values replace mistuning values forblades of the randomly-generated mistuningpattern. The study of the influence of a smallnumber of blades with very different frequencieswas undertaken to assess the following aspects:• Assess the damage tolerance of the assembly,
assuming a damaged blade would havedifferent dynamic characteristics to standardblades.
• What effect would introducing blades that maybe outside normal acceptance limits have andcould they be used without adverse effects? Ifthe effect was relatively benign then it may bepossible to extend acceptance limits andtherefore reduce costs.
Excitation by 6th, 7th and 8th engine orders (EO)was studied in the frequency range correspondingto the 1F mode, and excitation by 40th EO wasstudied in the frequency ranges corresponding tothe 1T mode. Loading for each blade wasdistributed uniformly over the blade nodes and thedamping loss factor was assumed equal 0.003.Excitation frequency ranges for each of EOsanalysed, natural frequencies of the tuned assemblyand mean values for experimental naturalfrequencies are shown in Fig.4. The frequencyvalues were normalised with respect to the firstblade-alone natural frequency, i.e. actual valueswere divided by a value of the first blade-alonenatural frequency.
4.2 An estimate of mistuning effect onresponse level and amplitude distribution
An example of the calculated amplitudes asfunctions of excitation frequency for all blades ofthe bladed disc is shown in Fig.5 for the case ofexcitation by 6EO. Fig.5a shows the bladeamplitudes for the case of a tuned bladed disc;Fig.5b illustrates the blade amplitudes for randomlymistuned bladed disc within the normal mistuning
7
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Fig. 4 Natural frequency of the tuned assembly andanalysed excitation engine orders, excitationfrequency ranges (red lines) and mean values forexperimental natural frequencies (blue lines)
range; and Fig.5c contains the blade amplitudes ofthe bladed disc having 90 randomly mistunedblades plus two damaged blades (namely, the firstand fifteenth blades). All amplitudes are normalisedby a value of the maximum amplitude of the tunedassembly. As one can see, the tuned bladed disc hasthe same amplitudes for all blades and there is onlyone resonance peak in the range considered – that isthe 6th nodal diameter mode. The introduction ofeven a small amount of mistuning leads tosignificant amplitude scatter over the blades of thebladed disc as well as to the appearance of a largenumber of resonance peaks. The highest peaks ofthe system with small mistuning lie near theresonance peak of the tuned assembly with thenormalised frequency value close to 1. Therandomly-mistuned bladed disc with the damagedblades has two additional resonance peaks in thevicinity of normalised frequency magnitude 1.35.Resonance frequencies corresponding to thesepeaks are close to the natural frequencies of theblade-alone damaged blades.
Envelopes of the maximum responses foundover all blades of the assembly are shown in Fig.6for all the considered excitation types. Here,maximum normalised amplitudes are compared forthe tuned bladed disc, for the randomly-mistunedassembly with a normal, small, mistuning level, andfor the randomly-mistuned assembly having twodamaged blades. For the case of 6EO excitation inthe range of the 1F mode, one can see that smallrandom mistuning gives rise to many resonancepeaks. These peaks lie in the vicinity of the singleresonance peak of the tuned system and far from itas well.
a)
b)
c)
Fig. 5 Forced response for each blade of the bladeddisc: a) tuned assembly; b) randomly mistunedblades; c) random mistuning and a damaged blade.The case of 6EO excitation in 1F mode frequencyrange.
The highest resonance peaks of the mistunedsystem are close to the resonance of the tunedassembly. Some of these peaks are higher than theresonance peak of the tuned system although manyof them are lower. Owing to the presence of manyresonances, the envelope of the maximum responsehas a much wider frequency range with highamplitudes than the one forthe tuned system. For
8
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Fig. 6 Comparison of envelopes of maximumnormalised response for tuned assembly, randomlymistuned one and the bladed disc containingdamaged blades.
the case of the bladed disc with the damagedblades, two new resonance peaks appear near thenatural frequencies of the damaged blades and alsonear other assembly modes in the mistuned system.For the case of excitation by 6EO maximumnormalised amplitude of such bladed disc has avalue which is much higher than worsening effectcaused by normal mistuning. The maximumresponse for the cases of excitation by 7EO and8EO also lead to similar conclusions about theinfluence of small mistuning and damaged bladesto forced response. The only difference is the muchlower levels of normalised amplitude (less than 1)for the resonance peaks in the vicinity of the naturalfrequencies of the damaged blades. Analysingexcitation by 40EO in the 1T frequency range inFig.6d, one can see that the introduction of smallrandom mistuning in this case leads to a largeincrease in the frequency band with largeamplitudes compared with that of the tuned system.This frequency range increase is essentially greaterthan for the case of excitation in the 1F modefrequency range. As is typical for mistunedassemblies, the maximum resonance peaks arehigher than those of the tuned bladed disc. Thedamaged blades for these cases do not change verymuch maximum response level and character of theenvelope of forced response.
Correspondence between distinctive resonancepeaks of the randomly-mistuned bladed disc andnatural frequencies of its tuned counterpart isdemonstrated in Fig.7. Here, six distinctiveresonance peaks are numbered in Fig.7a showingthe envelope of the response, and the naturalfrequencies of the tuned system correspondingthem are shown in Fig.7b. All these peaks are onsloping parts of curves in the natural frequencies-nodal diameters diagram, where blade vibrationcoupling through the disc is high. Many resonancepeaks corresponding to the natural frequencieslying on the flat parts of the first curve form arelatively wide normalised frequency range (from0.96 to 1.0) with high response levels. This range isformed by combined resonance peaks, which arealmost indistinguishable.
In Fig.8 distributions of amplitudes over all theblades of the bladed disc are displayed whichcorrespond to the first four of the distinctiveresonance peaks shown in Fig.7. One can see thatone or a small number of blades can have muchhigher amplitudes than the others for someresonances. For the 4th resonance peak analysed,the distributions of amplitudes have a regularpattern. This peak corresponds to the naturalfrequency of the tuned system, which is distantfrom any others.
9
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b) Fig. 7 Correspondence of the distinctive resonancepeaks for the mistuned assembly a) and naturalfrequencies (b) of the tuned one: 6EO excitation
4.3 Forced response analysis for randomly-generated mistuning patterns
A statistical survey of the worsening effect ofmistuning has been carried out for many randomlygenerated patterns. A small reduction of thefrequency ranges used for the case of statisticalcalculations was made to compare maximumamplitudes of the required modes (1F/6EO,1F/7EO, 1F/8EO, and 1T/40EO) only. Suchreductions in the frequency range were made afterperforming an FRF analysis, which showed highamplitudes near boundaries of the consideredfrequency ranges. These amplitudes weredetermined by interactions of a damaged bladevibrating in one mode and the bladed disc assemblyvibrating in another family of modes. The lattercase appeared for the bladed discs with damagedblades owing to their large frequency deviationsfrom the mean value for the whole bladepopulation.
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Fig. 8 Amplitude distribution over the bladed discat the distinctive resonances
The maximum, mean and minimum amplitudesfor each of the analysed mistuning patterns areshown in Figs.9-10. The maximum amplitude wasfound over all blades of the mistuned bladed discsand among all excitation frequencies from therange considered. The mean amplitude wasdetermined as the sum of the maximum amplitudesfound for each blade in the considered excitationfrequency range divided by the number of blades.The minimum amplitude was determined as thevalue minimum found from the maximumamplitudes found for each blade in the consideredrange.
10
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Fig. 9 The case of 6EO excitation in normalisedfrequency range 0.94…1.4. Maximum, mean andminimum amplitudes found over all blades for eachmistuning patterns analysed: a) randomly generatedmistuning patterns; b) randomly reshuffled bladesfrom a given set
In the first plot (a) of each of Figs.9-10, theamplitudes are given for the following types ofmistuning:1) random blade mistuning within normal
mistuning range (M1);2) one damaged blade and the rest blades are
randomly mistuned (M2);3) two damaged blades and the rest blades are
randomly mistuned (M3).In total, 25 different mistuning patterns weregenerated for each of these types. For the 2nd and3rd types the positions of the damaged blades werefixed while for the blades with normal mistuningrange, the actual frequency mistune pattern wasgenerated using characteristics of the fitted normaldistribution.
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Fig. 10 The case of 40EO excitation in normalisedfrequency range 3.2…4. Maximum, mean andminimum amplitudes found over all blades for eachmistuning patterns analysed: a) randomly generatedmistuning patterns; b) randomly reshuffled bladesfrom a given set
For the second plot in each of these figures,(b), the following types of mistuning areconsidered:1) two damaged blades and the rest are tuned,
positions of the damaged blades are randomlychanged (M4);
2) 90 experimentally-determined mistuned bladeswith 2 damaged blades which are reshuffledchanging positions of the all 92 mistunedblades in the bladed disc (M5).
For this plot random reshuffling of the initialmistuning pattern was performed (i.e. all 25 casesused the same set of blades, but rearranged indifferent order). To facilitate references the fivemistuning types considered are marked here bysymbols M1…M5.
11
For example one can see that for the 1F modeexcited by 6EO (Fig.9) the introduction of thedamaged blades into a randomly-mistunedassembly results in only small changes ofmaximum, mean and minimum amplitudes. Theeffect of damaged blades on maximum responselevels is more significant here for the case of bladereshuffling.
In Table 1 some statistical characteristics forall the analysed mistuning patterns are given.Namely, the mean values and scatter ranges over allthe analysed mistuning patterns are determined forthe above-mentioned maximum, mean andminimum amplitudes found for each mistuningpattern. One can see that the scatter of maximumamplitudes usually lies within the range ±30% ofthe average and for mistuning patterns obtained byblade reshuffling this scatter is smaller than for thecase of randomly-generated mistuned blade sets.The scatter is especially small for the case whenonly two blades are mistuned and their positions arechanged as a result of random reshuffling. Thescatter of mean amplitudes is small and is usuallymeasured in a few percent.
Table 1 Mean values and scatter of the normalisedresponse for all analysed mistuning patterns: 6EO
excitation
Maximumnormalisedresponse
Mean normalisedresponse
Type ofmistuning(see page
10)Mean for
allpatterns
Scatter,%
Mean forall
patterns
Scatter,%
M1 2.72 -20…28 0.65 -8…15
M2 2.74 -25…28 0.63 -8…19
M3 2.71 -20…25 0.65 -8…15
M4 2.87 -13…12 0.90 -1.3…1.9
M5 2.37 -15…23 0.70 -4.0…5.9
The maximum normalised amplitudes foundover all the analysed mistuning patterns are givenfor each kind of excitation Table 2. The worseningeffect of the mistuning is largest for the 1F/6EOresonance condition for the considered bladed discand excitation conditions. It results in more thanthree times higher amplitudes for some blades thanfor the tuned system. The least mistuning
worsening effect is for 1F/8EO and 1T/40EOexcitations, when the maximum normalisedamplitudes are around 2.3 and 1.8 respectively.Introduction of the damaged blades does not affectthe worsening much when frequency rangesincluding resonances determined by the requiredblade mode only (i.e. by 1F or 1T modes) areconsidered. However, it should be noticed (seeFig.6) that the damaged blades caused theappearance of large resonance peaks in the widenedfrequency ranges, where the damaged blade naturalfrequencies are close to natural frequencies of thebladed-disc assembly.
Table 2 Maximum normalised amplitude levelsfound over the all analysed mistuning patterns
Kind of excitationType ofmistuning(see page
10)1F/6EO 1F/7EO 1F/8EO 1T/40EO
M1 3.495 2.786 2.290 1.709
M2 3.493 2.773 2.163 1.651
M3 3.379 2.756 2.009 1.679
M4 3.211 2.710 1.457 1.512
M5 2.921 2.676 2.103 1.795
Total overall the types
3.495 2.786 2.290 1.795
5. Conclusions
An efficient method for the forced vibrationresponse analysis of mistuned bladed discs usingdetailed finite element models has been presented.The method proposed here allows the exactcalculation of the response of mistuned bladeddiscs using FRF matrices of the correspondingtuned system and together with a mistuning matrix.The distinct features of the proposed method are: (i)only a single sector model is needed to representthe tuned and mistuned bladed disc, (ii) mistuningis treated as a structural modification problem, (iii)the computational cost for mistuning calculationsdoes not depend on the size of the original sectormodel as the solution is obtained at active co-ordinates, and (iv) the reduced modelcorresponding to the active co-ordinates is asaccurate as the initial sector model represented byits natural frequencies and mode shapes.
12
Investigations of the forced response ofmistuned bladed discs containing randomlymistuned blades (within the normal frequencymistuning range), and with damaged blades, havebeen carried out. The worsening effect of mistuningon resonance amplitude levels and on forcedresponse functions has been analysed usingdetailed, realistic finite element model. The highestamplitude level was determined from manymistuning patterns. These mistuning patterns werecreated by: (i) generation frequency mistuning foreach individual blade randomly; (ii) randomreshuffling a predefined mistuning set.
6. Acknowledgement
The authors are grateful to Rolls-Royce plc. forproviding the financial and technical support forthis project and for giving permission to publishthis work.
7. References
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of the IUTAM Symposium: "The Active Control ofVibration," Bath, UK, pp. 189-196?�A Petrov, E.P., Sanliturk, K.Y., Ewins, D.J., Anew method for dynamic analysis of mistunedbladed discs based on exact relationship betweentuned and mistuned systems. (submitted to Trans.ASME: J. of Eng. for Gas Turbines and Power)?��A Sanliturk, K. Y., Ewins D. J. Elliott R andGreen J.S, 1999, “Friction damper optimisation:Simulation of rainbow tests” Presented at theASME TURBO EXPO’99, Indianapolis, USA.ASME Paper 99-GT-336. (Also accepted forpublication in the Transactions of the ASME).?��A Slater, J.C, Minkiewicz G.R., and BlairA.J., 1999, ”Forced response of bladed diskassemblies – A survey,” The Shock and VibrationDigest, Vol. 31, No 1?��A Sherman, J., and Morrison, W.J., 1949,“Adjustment of an inverse matrix corresponding tochanges in the elements of a given column or agiven row of the original matrix,” Ann. Math.Statist., 20, p. 621.?��A Whitehead, D.S., 1966, “Effect ofmistuning on the vibration of turbomachinesinduced by wakes,” J. of Mechanical EngineeringScience, Vol. 8, No. 1, pp. 15-21?��A Woodbury, M. , 1950, “Inverting modifiedmatrices,” Memorandum Report 42, StatisticalResearch Group, Princeton University, Princeton,NJ?��A Yang, M.-T., and Griffin, J.H., 1999, “Areduced order model of mistuning using a subset ofnominal system modes,” Presented at the ASMETURBO EXPO’99, Indianapolis, USA. ASMEPaper 99-GT-288.